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Biswajit Mitra

Publications and source records attributed to Biswajit Mitra.

10 recordsLinked to original sources
arXiv2026

Spectral Homotopy and the Spectral Fundamental Group

In this paper, we introduce an algebraic-topological invariant for commutative pm-rings, termed the spectral fundamental group, which is denoted by $π_{k}^{alg}(A)$. This group is defined via homotopy classes of loops within the space of induced spectral maps, which are generated by the $k$-algebra endomorphism monoid of the ring. We establish foundational properties of this invariant, proving that $π_{k}^{alg}(A)$ is an abelian group that naturally respects direct products and admits natural morphisms with respect to fully invariant subrings. Further, we establish an explicit isomorphism between the spectral fundamental group of certain continuous function rings and the classical fundamental group of their associated topological mapping spaces. Finally, utilizing a generalized dual number construction, we present an explicit example of a pm-ring that cannot be embedded into any function ring over a field of characteristic zero, yet possesses a nontrivial spectral fundamental group. This demonstrates that $π_{k}^{alg}(A)$ captures homotopical dynamics that are intrinsically algebraic.

math.AT↗
arXiv2025

On semi-transitional and transitional rings

In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general framework for these rings is developed through the notion of semi transition and transition maps, leading to a systematic exploration of their algebraic and topological properties. Structural results concerning product rings, localizations, and pm rings are established, showing that these new classes naturally generalize familiar examples such as polynomial rings over fields, rings of bounded continuous functions, and the ring of admissible ideal convergent real sequences. Ideals and filters induced by semi transition maps are analyzed to characterize prime and maximal ideals, revealing a duality between algebraic and set-theoretic constructions. Furthermore, conditions under which semi transitional rings become semiprimitive are determined, and a Stone Cech like compactification is constructed for transitional rings, giving rise to a new perspective on unique extension properties in topological algebra.

math.AC↗
arXiv2024

Essential subgroups and essential extensions

The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated. Further, we have completely characterized groups that do not possess a proper essential extension. These observations are used in concluding several properties of groups having essential subgroups. Finally, a short proof of the well-known theorem of Eilenberg and Moore that the only injective object in the category of groups is the trivial group is given.

math.GR↗
arXiv2022

Socle of Hamiltonian Group

The socle of a group $G$ is the subgroup generated by all minimal normal subgroups of $G$. In this short note, we determine the socle of a Hamiltonian group explicitly.

math.GR↗
arXiv2022

Few remarks on essential modules

In the present paper, modules over integral domains and principal ideal domains that are proper essential extensions of some submodules are classified. We introduce a new class of modules that we call $\mathrm{SM}$ modules and show that the class of Artinian modules, locally finite modules, and modules of finite lengths are all proper subclasses of SM modules. We also show that non-semisimple $\mathrm{SM}$ modules possess essential socles. Further, we show that non-semieimple modules over integral domains with nonempty torsion-free parts do not possess essential socles.

math.AC↗
arXiv2022

Studies of certain classes of functions and its connection with $S$-embeddedness

We call a function $f$ in $C(X)$ to be hard-bounded if $f$ is bounded on every hard subset, a special kind of closed subset, of $X$. We call a subset $T$ of $X$ to be $S$-embedded if every hard-bounded continuous function of $T$ can be continuously extended upto $X$. Every $S$-embedded subset is $C^*$-embedded. In this paper we have given a characterization of the converse part. To get the converse, we came across a type of function which are bounded away from zero on every hard subset of a subset. We further studied few properties of this type of functions and also of hard-bounded functions.

math.GN↗
arXiv2022

C(X) determines X -- an inherent theory

One of the fundamental problem in rings of continuous function is to extract those spaces for which C(X) determines X, that is to investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y . The development started back from Tychono? who first pointed out inevitability of Tychono? space in this category of problem. Later S.Banach and M. Stone proved independently with slight variance, that if X is compact Hausdor? space, C(X) also determine X. Their works were maximally extended by E. Hewitt by introducing realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop an inherent theory of this problem to cover up all the works in the literature introducing a notion so called P-compact spaces.

math.GN↗
arXiv2022

Construction of nearly pseudocompactification

A space is nearly pseudocompact if and only if $\upsilon X\backslash X$ is dense in $βX\backslash X$. If we denote $K=cl_{βX}(\upsilon X\backslash X)$, then $δX=X\cup(βX\backslash K)$ is referred by Henriksen and Rayburn \cite{hr80} as nearly pseudocompact extension of $X$. Henriksen and Rayburn studied the nearly pseudocompact extension using different properties of $βX$. In this paper our main motivation is to construct nearly pseudocompact extension of $X$ independently and not using any kind of extension property of $βX$. An alternative construction of $βX$ is made by taking the family of all $z$-ultrafilters on $X$ and then topologized in a suitable way. In this paper we also adopted the similar idea of constructing the $δX$ from the scratch, taking the collection of all $z$-ultrafilters on $X$ of some kind, called $hz$-ultrafilters, together with fixed $z$-ultrafilter and then be topologized in the similar way what we do in the construction of $βX.$ We have further shown that the extension $δX$ is unique with respect to certain properties.

math.GN↗
arXiv2022

Study of spectrum of certain subrings of a commutative ring with identity

By a ring we always mean a commutative ring with identity. It is well known that maximal spectrum of $C(X)$, $C^*(X)$ and any intermediate subrings between $C(X)$ and $C^* (X)$ are homeomorphic and homeomorphic with $βX$, the Stone-$\check{C}$ech compactification of $X$. In this paper we generalized these results to an arbitrary ring by introducing a notion of dense subring. We proved that if $A$ is completely normal and dense subring of $B$, then maximal spectrum of $A$ and $B$ are homeomorphic and hence maximal spectrum of all intermediate subrings between $A$ and $B$ where $A$ is dense, are homeomorphic. We also proved that $A$ is dense subring of $B$ if and only if spectrum of $B$ is densely embedded in spectrum of $A$ and have further shown that if $A$ is dense subring of $B$, any minimal prime ideal of $A$ is precisely of the form $Q\cap A$ for some unique minimal prime ideal $Q$ of $B$. As a consequence, we concluded that if $A$ is dense in $B$, minimal spectrum of $A$ and that of $B$ are homeomorphic. We also studied different properties of dense subrings of a ring.

math.GN↗